# Wiegner Eckart theorem, I don't understand it!

## Main Question or Discussion Point

Hello,
In derivation of quadrupole effect (which influences NMR spectrum for nuclei with spin ≥ 1/2), there is one step I do not understand, it is Eckart Wiegner theorem, more specific:

just relavant equations:

Qαβ=∫(3xαxβαβr2)ρdr

So here is the W-E:

<I,m|Qαβ|I,m'>=C<I,m|3/2(IαIβ-IβIα)-δαβI2|I,m'>

and we expres constant C witm matrix element for m=m'=I, and α=β=z:

eQ=<I,I|QzzI,I|>=C<I,I|3Iz2-I2|I,I>=<I,I|I(2I-1)|I,I>

questions:
how can we express Qαβ with spin operator?
can someone comment on every step what is being done? I would like to understand this intuitively if possible...Great thanks!

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Bill_K
Jamalll, What the Wigner-Eckart Theorem says is that the matrix elements of all quadrupole operators have the same m dependence.

Not saying that Q has any simple relationship to I, it's just that 3/2(IαIβ + IβIα) - δαβ is the easiest quadrupole operator to construct and evaluate, and so we use it on the right-hand side to give us something to compare the matrix elements of Q with.

But I still can not make any sense out of it.... I know I repeat myself, but
how do you "convert" coordinates in spin operator? What is the significance of "m" and" m' "?

I know it is the state of nucleus with spin "I" and spin projection "m", right?

Well I am sorry but there is not a single mind which can desolve this?

Baby I am only human!!!